Logical form or syntactic structure does not have to issue from inborn powers in our brains, nor does it have to come from a priori structures of the mind. It arises through an enhancement of perception, a lifting of perception into thought, by a new way of making things present to us

(Robert Sokolowski - The Phenomenology of the Human Person)

Nicely said. We watch and learn.

IMO logic arose from observing the relationships between physical entities. Cause and effect and so on. But I admit to being very shallow on the origins of logic, and avoid "being"and other vague concepts.

This is the same problem with space as there may be with time.
The Planck length is the smallest unit of length, approximately equal to 1.616 x 10^-35 meters.
In a sense, our measurements of both space and time may be fundamentally flawed, in that, as there is no position in space, there may be no moment in time. — RussellA

Actually, the Stoney scale is roughly 1/10 the duration of the Planck scale in time measurements. These are simply units of measurements, not ultimate bounds in a some philosophical sense. If you were to ride on that photon as it traverses a Planck length, time would vanish completely for you.

I have wondered why certain physical facts about time have not entered into these discussions. For example, as you stand on the side of the road and someone passes you in a car going 60mph, time is measured infinitesimally slower for them compared to you. And someone drifting above you in a balloon measures time infinitesimally faster than you. In the calculus of physics an instant of time is a limit concept, not an established fact.

As a math prof I would give them a short written quiz. Or, I would discuss the topic with them, one on one, and discover how well they understood it.

This might not be appropriate in polite society. :cool:

It's how we are raised. As a child my family moved from city to city every two to three years, so I became conditioned to leave the past behind. One place for three years to graduate from high school, then college two years here then two years there, USAF two years and two years, grad school, teaching, etc. before finally moving in middle age to the house where I have resided for forty years.

Nostalgia is a distant concept. As is identity linked to a particular place or dwelling. Forgive me, but I think I will avoid Kundera.

Jan, hello and welcome. The need or exploitation of details depends upon the topic. Discussions of the classical notions of "being" may veer off into assorted details since the subject is not well defined, but when occasionally a topic in mathematics is introduced the necessity for details arises due to the subject's highly defined structure, whether one agrees with or disputes accepted qualities.

Can you tell I was a mathematician?

This seems to suggest the only reliable description of time requires a conscious observer, right? — Hanover

Just nit picking, but "reliable description" implies conscious observers. If no observers then reliable description is meaningless.

I'll get around to watching these. I assume Wolfram is still promoting his ideas about Cellular Automata as a foundation for science and math. I bought his enormous book when it first came out, but flamed out of reading it after several hundred of the thousand pages or so. It did encourage me to write several computer programs and experiment developing the patterns he alludes to.

It became a joke at scientific cocktail parties that almost everyone had a copy, but no one had completed reading it.

Thanks for bringing to my attention. But I must admit, going on 89, that having turned off the sound I fell asleep part way through. It seems like it's a discussion about the origins of calculus where the limit definition is given in terms of infinitesimals, called monads or whatever by Newton and Leibnitz. Weierstrass and Cauchy improved upon it by introducing epsilons and deltas.

Non-standard analysis is the modern version, but is not a popular approach in universities.

Symmetry (function)

↪jgill
how do you know is the same distance — Danileo

23 steps out, 23 steps back.

Comes from

${{e}^{i\theta }}=\cos (\theta )+i\sin (\theta )$
and
${{e}^{z}}=1+\tfrac{z}{1!}+\tfrac{{{z}^{2}}}{2!}+\cdots$

Is a poetic way of relating the important numbers of math, pi, e, i, 1, and 0.

↪unenlightened
yea approximate symmetry rules the world, but pure symmetry is not, as everything is affected by particles symmetry is impossible — Danileo

In mathematics, metric spaces are sets of points and a measure of distance between them: d(x,y)
such that d(x,y)=d(y,x), symmetry. In the real world, is the distance between my front door and my mailbox the same as the distance between my mailbox and front door?

↪jgill
Not sure i followed that, but isn't it also correct that the length of the "hole" is zero? — Hanover

The "length" of any number on R is zero. Numbers are positions on the real line, designated as points, none of which have any "length".

(To see if you comprehend what I say, Counselor, you might show that the interval [0,1] is in 1:1 correspondence with the interval [0,1). But don't worry about it.)

There is a nice mathematical way to cash our the intuition the original poster is gesturing towards. See The continuum as a final coalgebra shows that the real numbers (a.k.a. the continuum) can be constructed from infinite steaming interactions over infinite sequences of natural numbers. — FirecrystalScribe

Category theory. Beyond my grasp and interest. Nevertheless, what is an "infinite steaming interaction"?

Is this the same as saying that the infinity of all integers is larger than the infinity of all even integers? Or, is it the same as saying that that is you have two sets, one composed of all numbers and the other composed of all numbers except the number 3, the first set is larger than the second?

In my first question, both sets are countable.

In my second question, neither are countable because both contain irrational numbers — Hanover

Regarding question 2, consider two horizontal non-negative real lines, one above the other. The line on top is missing the number 3. Do they have the same cardinality? That means a one-to-one correspondence between the two lines. Start at the bottom line at 0 <-> 0 on the top line. Let x be on bottom line and y on top line. Then x=y until the "hole" where y=3. You have a 3 on the bottom, but not on the top. So you introduce the correspondence 3 <-> 4, then it's as usual, x=y, until you reach 4 on top, which has been taken. So, stipulate 4 <-> 5. Since this pattern persists indefinitely the one-to-one correspondence holds forever.

I've made the comment on several occasions that the Schrödinger equation, in its simplest form, is a concept from elementary calculus: the instantaneous rate of change of something is proportional to the amount of that thing at that instant. In calculus the solution of such an equation involves the exponential function, and in the setting of complex analysis, e^it = cos(t)+isin(t), which is a wave structure. Not a physical wave. Nice video, thanks.

Does my complicated odyssey end should I go deep enough within myself to open the door of my inner Conscience, then find a way to step into my soul? — jufa

I like your poetic style, but unless you describe this inner progression in at least minimal detail and elaborate on "complicated odyssey" the reader is left with nothing but style.

An experiment I read of some time back involved a person reacting to a warning signal without actually being conscious of it. If you are reading a book you are aware of the act, though your thoughts are focused on what you are reading.

Sounds like the Peace Corps here in the USA.

In mathematics, a theory has substance when it is deemed important or significant in some way by a community of scholars.

First the commitment to non-contradiction, to both avoid contradicting oneself in belief — boethius

Do you mean blocking the ability to see both sides of an issue? Give some examples please. I don't read lengthy essays.

But when scientists go beyond compiling facts to explaining their significance, they are straying into metaphysics, and doing Philosophy — Gnomon

Not if they speculate within the normal scope of science. But, if they conjecture that action at a distance has religious connotations, or that the universe is a reification of mathematics, then, yes.

. . . but everytime you open a banana, you are the first person to ever see it. — Hanover

How exciting. Be sure and post its image on TPF. :cool:

↪Quk
I've never seen anything uncaused. I have no reason to think that would fail prior to the big bang. Maybe a better thing would be to say "I want to know why the singularity existed" — AmadeusD

Good point. Years ago I published a mathematical result that, more or less in this context and under certain conditions, could be interpreted showing that the further back in time one goes from a current event the less it matters what the starting point is. This assumes no boundaries on what lengths the causal chain extends backward.

The Big Bang seems a bit like an essential singularity in complex analysis, as does a black hole. Absolutely bizarre things happen in its vicinity.

Howz bout dis'...if it ain't yer original thought, or yo' original thought about somebody else's thought, don't post? — alleybear

I wonder how many original thoughts have been posted on this forum? Fifty years ago my math advisor said, "You might think you have an original thought, but it's likely someone has had that thought before, perhaps in another context".

I believe the "American Dream" is more along the lines of starting life with little and making a financial success and/or gaining recognition later on. I have been privileged to know someone who started out almost penniless in his early twenties (I once went with him searching for stovepipe in trash dumps) , and became a billionaire, and was on an advisory panel with Bill Clinton in the 1990s. And I knew another of that generation who started from poverty and eventually created a clothing line and was worth hundreds of millions of dollars when he passed a few years ago.

Are there other nations where this kind of Dream is possible?

Horatio Alger personified and wrote of this in the 1800s.

Yet then it's typical that the limit approaches infinity — ssu

"x goes to positive infinity" simply means "positive x increases without bound". No need for the word "infinity". Simply shorthand. But, set theory is another matter and postulates all sorts of things.

I belong to a past generation who didn't venture beyond the Peano axioms. These days searching for relationships between the broad spectrum of math subjects is in fashion. I cheer them on from the sidelines. :cool:

There wouldn't be this kind of over and over repeating debate Zeno's paradoxes, if we fully would understand the infinite or infinity — ssu

Or the limit concept.

OK, if we have countable and uncountable infinities, what is the relationship between these two infinite sets? — ssu

Take the rationals in [0,1] and form the union with the irrationals in [0,1] and you get a continuum. They are complementary in the complete interval - which itself is a complete metric space with the usual metric.

The precision of position and momentum are proportional to each other such that a greater precision of position results in a lesser precision of momentum. — Moliere

How does this enter into a discussion of these Zeno type paradoxes? Define the momentum of a point as it progresses to zero. Does the tortoise have momentum? Too much of a stretch for me.

Is this in any way motivated by the uncertainty principle? — Moliere

Actual measurements fail beyond Planck's constants. These paradoxes are all hypothetical involving motions of dimensionless points along rational number scales.

(The proof of a limit is intensional, whereas the empirical concept of motion is extensional). — sime

Heuristics may not be precise, but its value can be substantial in an introductory course. I've never come across this sort of philosophical detail in elementary calculus, which includes lots of motion. "f(x) approaches . . . as x approaches . . ." is common language in math. But, whatever you say.

Hence calculus does not say that f(x) moves towards L as x moves towards p. — sime

Rubbish. :roll:

But also homeownership, which I kind of “fell for” in a way. Or the way living at home with one’s parents is viewed as being a loser — which ties into encouraging owning a home or renting. — Mikie

And what's the alternative? Are you implying that owning a home is a bad thing? Or renting?

I think it's fair to say that 'field' is used in many contexts: different disciplines in science and the humanities are commonly referred to as fields — Janus

Even mathematics is a little sloppy in this regard. A vector field is not a mathematical field. The reason I prefer the expression vector space. And if that vector space changes values at each point over time it is a time dependent vector space (or field).

So the question for you is, does every point in that field have a mathematical description, as do the points within physical fields? And if not, does that disqualify its description as ‘a field’? — Wayfarer

Field as a mathematical term or field as an area of land devoted to growing crops? Or field as an encompassing environment of some sort, a philosophical notion. Spacetime is not a math field, but contains various entities like magnetic fields that can be represented as math fields.

"Understanding" quantum theory means following the math, as Feynman said. Perhaps that is true of time as well. The math of relativity theory weaves an astounding vision far beyond what we might have imagined. If one entertains Tegmark's speculations that the universe is a mathematical structure, then time is one also. A reification of mathematics.

. . . excitation of the one universal field of subjectivity. — Wayfarer

From math to woo. A little like the aether.

As for understanding space/time, my Corgi still cannot comprehend simple high school algebra. We have to learn our limitations.

Are you recommending adding this to a classical education?